Method, apparatus and computer program providing broadband preconditioning based on reduced coupling for numerical solvers

ABSTRACT

This invention relates to computing numerical solutions of linear systems of equations, specifically to implementing preconditioning of the coefficient matrix of such a system. The preconditioning applies to any coefficient matrix, dense or sparse, based on the solutions of a physical problem of unknown functions, commonly referred to as basis or interpolation functions, where the basis function spans more then one mesh element. Examples of such linear systems can result from, as examples, an electromagnetic analysis of printed circuit boards or field scattering in radar applications, fluid mechanics and acoustics. A method and system to compute a preconditioner for a coefficient matrix A that is compatible with the linear system of equations that provides basis function support over at least two mesh elements. Coupling of the preconditioner between partitions of a portioned mesh representation is only through basis functions at the partition boundaries.

TECHNICAL FIELD

This invention relates to techniques and systems to compute numericalsolutions of linear systems of equations, and more specifically relatesto implementing preconditioning of the coefficient matrix of such asystem.

BACKGROUND

Systems are known in the art for computing numerical solutions of linearsystems of equations. Also known is the preconditioning of thecoefficient matrix of such a system.

Computer aided design (CAD) systems have become accepted universally asimportant engineering design tools for a variety of applications inmechanical, civil, and electrical engineering. Information regarding theshape and physical properties of a device or structure are captured andstored in the system in the form of a geometric model. The behavior ofthe physical system is described in mathematical terms, via theapplication of known laws of physics, as a problem whose solutionreveals a property of the system that it is desired to evaluate. Inpractice, a region is divided into elements, often tens of thousands ormillions of elements, over which the properties are considered to varyin a known manner and to which classical principles can be applied toyield a system of linear algebraic equations.

The computational cost associated with evaluating the physical system isdominated by the solution of this system of equations. It is thereforeof paramount technical importance to improve the computationalefficiency with which the numerical solution to these equations can begenerated. For this reason, iterative techniques are typically themethod of choice for solving such linear systems due to their speedadvantage over direct inversion methods. These methods, however, arestrongly dependent of the condition number, i.e., the size of the spanof the spectrum of the resulting matrix. It is known that largecondition numbers reduce the convergence of the system dramatically.

In order to overcome the convergence rate difficulties, matrixpreconditioners are used to reduce the condition number. Solving linearsystems using conjugate gradient (CG) iterative methods, for example,with preconditioning has been well documented and is summarized in“Numerical Recipes”, W. Press et al., Cambridge Press (1992). The mostcommon preconditioning method is known as incomplete LU, and isdescribed in detail in “Iterative Solution Methods”, O. Axelsson,Cambridge Press (1994). To summarize, in a triangular (LU) factorizationof a matrix A, the matrix is decomposed into a lower triangular matrix Land an upper triangular matrix U, thereby expressing the matrix as aproduct LU=A. If A is a sparse matrix, meaning the matrix has a lowratio of nonzero elements, the ratio of nonzero elements in theresulting matrices L and U is increased. The generation of nonzeroelements is ordinarily called “fill-in”. In the incomplete LU, thefill-in is ignored by approximating it to zero. Alternatively, fill-incan be selectively ignored based on its relative magnitude.Modifications to the incomplete LU in generating preconditioners forsparse coefficient matrices are described in U.S. Pat. No. 5,136,538,“Preconditioned Conjugate Gradient System”, N.K. Karmardar et al., andin commonly assigned U.S. Pat. No.: 5,754,181, “Computer Aided DesignSystem”, V. Amdursky et al.

The application of incomplete LU is limited to instances where thecondition number is small, which is especially the case in the solutionof dense matrices, such as those produced when attempting to solveintegral equations. In order to apply an incomplete LU on a densepreconditioner, the preconditioner must first be “sparsed”. For example,setting to zero all matrix elements whose magnitude is below a certainthreshold, or setting to zero all elements in the preconditionerrepresenting interactions between elements in the physical systemseparated by distances larger then a certain threshold. All knownchoices of sparsing the preconditioner, including the two mentionedabove, when combined with incomplete, or even complete, LU factorizationproduce poor results when the condition number of A is large. Therefore,in order to overcome the convergence rate difficulty in general linearsystems with large conditioner numbers, a general and robustpreconditioning method is needed. Prior to this invention, this need wasnot adequately met.

A particular problem arises during the electrical analysis of devicesthat are small with respect to the wavelength, such as electronicpackaging structures, chips, interconnects, and printed circuit boards.The analysis of such structures results in linear systems of equationsto be solved, which are desired to be solved iteratively by a fast, lowstorage requirement solver such as the Fast Multipole Method or thepre-corrected FFT method. The use of an iterative solver requires thatthe condition number of the linear system (the ratio of the largest tosmallest eigenvalue) be as close to one as possible. When the conditionnumber is too large a preconditioner is used to condition the system andreduce the condition number. However, for such structures as thosementioned above, the use of common rooftop basis functions (also knownas Rao-Wilton Glisson or RWG functions) produces linear systems withlarge condition numbers such that no known preconditioner has beenavailable to properly condition the system.

As was noted above, the most common attempt at a preconditioner is onebased on geometrically close interactions. To form such apreconditioner, one sets to zero all interactions in the linear systemthat are geometrically separated by a defined distance. Such apreconditioner will only be effective, however, for structures that arelarge compared to the wavelength when the common rooftop or RWG basisfunctions are used. One existing solution to this problem uses lessdesirable loop-tree or loop-star basis functions which result in abetter conditioned linear system. A disadvantage of this approach,however, is that the fast solvers mentioned above have difficultiesdealing with the often large loops produced by the change in the basisfunctions. This is due to the inaccurate calculation of nearinteractions by the fast solver, for which large loops have many.Reference can be made to “Fast and efficient algorithms inelectromagnetics”, by Weng Chew et al., Artech House 2001, fordescriptions of fast solvers and loop-tree or loop-star basis functions.

SUMMARY OF THE PREFERRED EMBODIMENTS

The foregoing and other problems are overcome, and other advantages arerealized, in accordance with the presently preferred embodiments ofthese teachings.

Disclosed is a preconditioning method for the solution of linear systemsbased on physical systems, where unknown basis functions are supportedby at least two mesh elements. To generate the preconditioner, thephysical system is partitioned into regions with a small number ofunknown elements per region. The preconditioner is represented by thoseelements of the linear system's coefficient matrix representinginteractions between physical elements in the same regions only.

To avoid the use of any form of incomplete LU, which produces inadequateresults, a specialized method of applying the preconditioner is alsodisclosed. The preconditioning technique can be applied to any iterativesolver, such as conjugate gradient (CG) or the method of geometricresiduals (GemRes) and to any type of numerical solver, such as themethod of moments (MoM) or the finite element method (FEM). Thepreconditioning technique effectively lowers the condition number of illconditioned linear systems previously thought to be unsolvable in theirnative form due to their large condition number.

Disclosed is a method, a computer program and a system to partition themesh of a physical system into regions to produce a preconditionercompatible with linear systems of equations having basis/interpolationfunction support over at least two mesh elements. Coupling of thepreconditioner between regions or partitions is only through the basisfunctions on the partition boundaries, which produces a preconditionerthat is itself a valid solution to the same set of physical equationsgoverning the full linear system. In this way the preconditioner isideally suited for application to systems with poor condition numbers.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other aspects of these teachings are made more evidentin the following Detailed Description of the Preferred Embodiments, whenread in conjunction with the attached Drawing Figures, wherein:

FIG. 1 is a schematic drawing of a method to provide the solution of aphysical system.

FIG. 2 is a depiction of the splitting of a mesh into regions orpartitions for preconditioning.

FIG. 3 illustrates two reduced coupling partitions having a partitionboundary and shows, for a non-limiting example of an application of thisinvention, three types of currents (interior, across a boundary andalong the boundary).

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The preconditioning operation in accordance with this invention appliesto any coefficient matrix, dense or sparse, based on the solutions of aphysical problem of unknown functions, commonly referred to as basis orinterpolation functions, where the basis function spans more then onemesh element. Examples of such linear systems may result from, but arenot restricted to, an electromagnetic analysis of printed circuit boardsor field scattering in radar applications, fluid mechanics and acousticsusing solution techniques such as the method of moments (MoM) or thefinite element method (FEM). References for both methods can be found inR. Harrington's Field Computation by Moment Method, Krieger Publishing(1968) and J. Jin's The Finite Element Method in Electromagnetics, JohnWiley & Sons Inc. (2002), respectively.

FIG. 1 shows various operations for solving a linear system of equationsbased on the presently preferred method of preconditioning, that in turnis based on reduced coupling. FIG. 1 may also be viewed as representinga block diagram of a computational system that generates and uses thepreconditioner in accordance with this invention. FIG. 1 may also beviewed as a logic flow diagram that illustrates the operation of acomputer program having program instructions that are executable by atleast one digital processing apparatus, and that is stored in amachine-readable medium, such as in memory chips, a disk, a tape, or anysuitable computer program instruction storage media.

A presently preferred, but not exclusive, method of reduced coupling isdescribed by B. Rubin et al. in “Electrical modeling of extremely largepackages”, Proceedings of Electronic Components and TechnologyConference, 1997, the content of which is incorporated by referenceherein.

In accordance with an aspect of this invention the reduced coupling isapplied as a preconditioner to solve linear systems, and in accordancewith a further aspect of the invention the preconditioner may be appliedto a wider range of problem types than conventional approaches.

In the method of FIG. 1 a description 10 of a physical structure ofinterest is divided by a mesh generator 20 into smaller elements,potentially numbering anywhere from thousands to millions. The union ofthese elements is commonly referred to as the “mesh” (see FIG. 2). Inthe MoM approach, commonly used to solve integral equations, unknownvector functions are described by mathematically known functions calledbasis functions that are supported over adjacent mesh elements. In theFEM approach, commonly used to solve differential equations, basisfunctions are associated with the mesh vertices and are defined over allmesh elements using that vertex. The mesh is used to compute the matrixA of coefficients for the linear system in block 30 by computing theinteraction between the simple functions defined over all of these setsof mesh elements.

A preconditioning matrix K is computed by first splitting orpartitioning the mesh 20A that is output from mesh generator 20 intoindependent regions using mesh splitting function 40, with each regionor partition containing many elements. FIG. 2 depicts an example of aphysical mesh 20A split into five distinct partitions 21A, 21B, 21C, 21Dand 21E, collectively referred to as partitions 21. The partitionboundaries are constrained to coincide with the edges of mesh elements.In other words, the mesh elements may start and end on the partitionboundaries but may not cross the boundaries.

The preconditioner is implemented in block 50 by computing mesh elementinteractions using reduced coupling. The interactions between basisfunctions are computed only for those “half” functions within the samepreconditioning partition 21. Here a “half” denotes the function overany one of the multiple mesh elements for which it is defined. Theinteractions of basis functions crossing a partition boundary, labeledboundary elements, are computed separately for each of the halves. Thatmeans that no interactions exist between basis function halves that aredefined in separate ones of the partitions 21. Those basis functionscompletely within a partition 21, labeled interior elements, interactonly with other interior elements and boundary element halves in thesame partition 21. In this way, interactions in the preconditioningmatrix K between interior elements in the same partition 21 areequivalent to the interactions in the original matrix A, interactionsbetween boundary elements and interior/boundary elements in the samepartition(s) are different than the corresponding interactions in A.This is true due to neglecting some of the coupling between basisfunctions, and assuming that interactions between elements in differentpartitions are zero.

It is important to note that, unlike zeroing out interactions based ongeometric radius, as discussed previously, the resulting preconditioningmatrix is a valid solution to the original physical set of equations.More specifically, it is the superposition of several solutions, one foreach region. In this way it is ideally suited for preconditioning thematrix of the original system.

In accordance with the foregoing technique, the indices of the basisfunctions in the matrices A and K are sorted so that all internalelements appear first, grouped according to their respective regions,followed by all boundary elements. The resulting preconditioning matrixK for a geometry with n partitions 21 is computed in block 50 and is ofthe form:

$\begin{matrix}{{K = \begin{bmatrix}\lbrack {Ka}_{1} \rbrack & \; & \; & \; & \; \\\; & \lbrack {Ka}_{2} \rbrack & \; & \; & \; \\\; & \; & ⋰ & \; & \lbrack{Kb}\rbrack \\\; & \; & \; & \lbrack {Ka}_{n} \rbrack & \; \\\; & \lbrack{Kc}\rbrack & \; & \; & \lbrack{Kd}\rbrack\end{bmatrix}},} & (1)\end{matrix}$where the sub matrix Ka is the block diagonal matrix created by theunion of the matrices of internal element interactions Ka₁, throughKa_(n), Kd represents the interactions between the boundary elements,and Kb and Kc are the interactions between the internal and boundaryelements. In general, Kb and Kc are not square, and are sparse sinceeach column/row contains interactions between one boundary element andthe internal elements to the two regions sharing that boundary. Thematrix Kd is square, with dimension equal to the total number ofboundary elements, and is sparse, although its complete LU factorizationwill in general be dense.

In order to solve the system of equations Ax=f implied by the meshgenerator 20, the linear system matrix A from block 30, a vector f ofthe boundary conditions on each element from block 60, and thepreconditioner function K from block 50 are combined in an iterativesolver 70. The resulting output of the iterative solver 70 is theapproximate solution x80. The operation of the iterative solver 70, forexample CG, with the preconditioner function K proceeds as follows:

A trial solution X_(o) is selected. Vectors r_(o) and P_(o) are definedas r_(o)=f−Ax_(o) and P_(o)=K⁻¹r_(o). The following steps are thenperformed iteratively until convergence is achieved:α_(i)=(K ⁻¹ r _(i−1) ,r _(i−1))/p _(i−1) ,Ap _(i−1))x _(i) =x _(i−1)+α_(i) p _(i−1)r _(i) =r _(i−1)−α_(i) Ap _(i−1)β_(i)=(K ⁻¹ r _(i) r _(i))/(K ⁻¹ r _(i−1) ,r _(i−1))p _(i) =K ⁻¹ r _(i)+β_(i) p _(i−1)where the expression in the form of (x,y) represents the scalar productof the vectors x and y.

The application of the preconditioner function K requires the productw=K⁻¹v, where v and w are vectors. Since the inverse K⁻¹ would have alarger ratio of nonzero elements then K, this is performed by solvingthe system Kw=v. This is done by noting the division of K, w, and v intotheir internal and boundary elements:

$\begin{matrix}{\begin{bmatrix}v_{int} \\v_{bnd}\end{bmatrix} = {{\begin{bmatrix}{Ka} & {Kb} \\{Kc} & {Kd}\end{bmatrix}\begin{bmatrix}w_{int} \\w_{bnd}\end{bmatrix}}.}} & (2)\end{matrix}$

Using linear algebra operations, the solution for the boundary elementsw_(bnd) can be expressed as the solution of[Kd−KcKa ⁻¹ Kb]w _(bnd) =v _(bnd) −KcKa ⁻¹v_(int).  (3)

This requires only the LU factorization of the block diagonal matrix Kaand the factorization of the matrix [Kd−KcKa⁻¹Kb], which is of dimensionequal to the total number of boundary elements and is typicallyextremely small compared to the total number of current elements. Thesolution of the internal elements w_(int) is thenw _(int) =Ka ⁻¹ v _(int) −Ka ⁻¹ Kbw _(bnd).  (4)

The solution time for this linear system is dominated by the LUfactorization of the block diagonal matrix Ka, and assuming a constantnumber of unknowns per partition for simplicity, isO(N_(partitions)N_(elements) _(—) _(per) _(—) _(partitions) ³).Additionally, if the number of unknowns per partition is held fixed, andthe size of the problem is increased, thus increasing the number ofpartitions, then the CPU time and storage requirements grow as O(N),even for an originally dense matrix A.

To perform as much up front computation as possible before starting theCG, the LU factorizations of Ka and [Kd−KcKa⁻¹Kb] are preferably takenprior to starting the iterations. The traditional back substitution ateach iteration is replaced with a combination of the back substitutionsof the stored LU factorization of Ka, which is done as independent backsubstitutions on each block of Ka for maximum efficiency, twomatrix-vector multiplications with the sparse matrixes Kb and Kc, and aback substitution on the matrix [Kd−KcKa⁻¹Kb], which is dense but smallin dimension. The time spent preconditioning the system is dominated bythe back substitutions, but the required CPU time is O(N).

Based on the foregoing description it can be appreciated that thepreconditioner 50 partitions the description of the physical structureor entity, actually the mesh representation of the physical structure orentity, into isolated regions. Coupling between the regions orpartitions exists only in the form of basis functions which cross thepartition boundaries. The result is a preconditioner function K that isa superposition of smaller, valid solutions of the original set ofgoverning equations. For this reason, it is applicable to linear systemswith extremely poor condition numbers (>10¹¹). The storage requirementsof the preconditioner 50 are sparse, and require but O(N) operations (O)per iteration to apply, where N is the dimension of the original linearsystem.

Discussed now is an application of the foregoing teachings to aspecific, non-limiting class of problems, where the preconditioner 50allows full wave linear systems of both electrically small and largegeometries, with rooftop or RWG basis functions, to be properlypreconditioned.

As was discussed above, the preconditioner 50 is based on the reducedcoupling structure concept, the details of which can be found in“Electrical modeling of extremely large packages” by B. Rubin et al., inthe Proceedings of the Electronic Components and Technology Conference1997.

As was also discussed above, the original structure is partitioned intoat least two smaller structures, as shown in FIG. 2. Assume for thiscase that the original structure represents a signal line embodied as athick conductive loop 23. It is assumed that full electromagneticcoupling exists between electric and magnetic currents within eachpartition 21, and that the electric and magnetic currents do not couplewith other currents outside of their respective partition 21. Currentflow is enforced to be continuous across the boundaries of allpartitions 21. The matrix for the linear system that results from thisanalysis of the partitioned structure, labeled as the reduced couplingsystem matrix, is the preconditioner for the original structure. Thepreconditioner and the necessary LU factorizations are sparse, so thatthe storage requirements are small and the application is fast.

Linear systems to be preconditioned may include the system of theoriginal structure, such as the thick conductive loop 23, or a reducedcoupling system composed of larger partitions than the preconditioner.Although the preconditioner removes the requirement of loop-tree orloop-star basis functions, it could also be applied to condition linearsystems composed from these basis functions as well.

The generation of the partitions 21 could be automated, as shown in FIG.1 at block 40. A simple example of automated partitions is to divide thestructure into sections along each of the three coordinate axis,generating a grid of boxes that partition the structure.

In accordance with this invention the original structure is partitionedinto at least two smaller structures, and those mesh cells that cross apartition boundary are split so that they start and stop on thepartition boundary. In this way all mesh cells are located completelywithin one partition 21. A basic strategy for implementing the reducedcoupling is to compute the basis/testing function (also known as anexpansion/weighting function) interactions by treating each ‘half’ ofthe basis and testing functions independently. Here a ‘half’ denotes thefunction over one of its two mesh cells.

Each half basis is assumed to radiate electric/magnetic field onlywithin the same partition, and thus interacts only with half testingfunctions within the same partition. Testing functions crossing apartition boundary, labeled boundary elements, are tested separately forthe two halves. The half of the testing function in each partition testsonly the field from basis functions in that partition. If the basisfunction in question is a boundary element, then only the half of thebasis in that partition is tested. This also applies to the self terms,or diagonal terms, of the matrix. Testing functions located completelywithin a partition, labeled interior elements, operate to test theelectric field from other interior elements and halves of boundaryelements in the same partition. By forcing both halves of a boundaryelement to share the same coefficient in the linear system solution, theconduction current is continuous across boundaries if the partitions 21.

To illustrate the foregoing principles, FIG. 3 shows two partitions Aand B for which the electric and/or magnetic current is the solutiondesired, labeled current elements, that exist within the partitions. Theexample current elements I_(int), I_(bnd) and I_(tan) (I internal,boundary and tangent, respectively, as shown in FIG. 3) are given toillustrate the three possible types of basis/testing functions.

The type of current is defined by the location relative to the partitionboundaries. The current element I_(bnd) represents a basis/testingfunction whose functional support crosses the boundary between the twopartitions labeled A and B. When formulating the linear system matrix,the half of I_(bnd) within partition A is tested over all of the testingfunction completely contained within partition A, and any halves of thetesting functions from other boundary elements within A. The half ofI_(bnd) within partition B does not radiate into A, and thus itscontribution is not tested within partition A. The reverse applies tothe half of I_(bnd) within partition B. Even the self term, or diagonalelement, of the matrix does not contain interactions between the twohalves of I_(bnd). Both halves of the boundary element are given thesame coefficient in the linear system solution, thus current is forcedto be continuous across the boundary between the partitions A and B

The current element I_(int) represents a basis function whose support iscomplete contained within a single partition 21. The current I_(int) istested over all testing functions located completely within the samepartition, as well as the halves of the boundary elements within thepartition. All other interactions with I_(int) are zero.

The current element I_(tan) represents a basis/testing function whosesupport lies directly on, and does not cross, a partition boundary. Thepartition to which I_(tan) belongs is thus undefined. In this caseI_(tan) is either assumed to be an internal element to one of the twopartitions, or duplicated to produce two internal current elements, eachinternal to one of the two partitions that share the boundary.Duplication appears to provide superior performance. If only half of thebasis/testing function is tangent to the boundary, then the other halfmust be internal to one partition, and I_(tan) is treated as in internalelement to this partition only.

The indices of the basis/testing functions in the linear system aresorted so that all internal elements appear grouped according to theirrespective partitions, and boundary elements are grouped. The resultinglinear system matrix for a geometry of n partitions then takes the formshown above in Eq. (1), the use of the preconditioner 50 proceeds as wasdiscussed above.

It should be noted that there are at least two issues that are notobvious in the above-cited Rubin et al. publication that are believed toinhibit the disclosed technique from being most effectively used as apreconditioner. The first is that the diagonal elements of the matrixcorresponding to the boundary elements are different in the reducedcoupling system and in the original linear system. This is not the casefor other preconditioning schemes, and would not be intuitive whenimplementing a preconditioner. The reason for this difference, asmentioned above, is that, unlike internal elements, the two halves ofthe boundary element do not couple with each other since they arelocated in separate partitions. If this detail is ignored, the reducedcoupling preconditioner will be less effective.

The second issue relates to the fact that only those current elementsthat cross a partition boundary can be treated as boundary elements.While it might intuitively appear that the more coupling that isincluded in the preconditioner the better, this is not true in thiscase. Treating elements that are otherwise internal to a partition asboundary elements, by allowing them to couple to more than onepartition, will result in a preconditioner that is less effective.Basically, the structure of the coupling as described above should beadhered to, or the preconditioner will be less effective.

The foregoing description has provided by way of exemplary andnon-limiting embodiments a full and informative description of the bestmethod and apparatus presently contemplated by the inventors forcarrying out the invention based on the current knowledge of theinventors. However, various modifications and adaptations may becomeapparent to those skilled in the relevant arts in view of the foregoingdescription, when read in conjunction with the accompanying drawings andthe appended claims. As but some examples, the use of other similar orequivalent iterative solvers 70, the use of sparse storage methods onthe components of the preconditioner, the use of alternate mesh elementsor expansion/interpolation functions, considering the elements of anytwo adjacent or nonadjacent partitions to belong to a single largerpartition, and the use of this invention for other applications (otherthan to solve current flow in electrical conductors) may be attempted bythose skilled in the art. However, all such and similar modifications ofthe teachings of this invention will still fall within the scope of thisinvention.

Furthermore, some of the features of the present invention could be usedto advantage without the corresponding use of other features. As such,the foregoing description should be considered as merely illustrative ofthe principles of the present invention, and not in limitation thereof.

1. A method to determine information indicative of at least one propertyof a physical entity by utilizing a linear system of equations torepresent the physical entity, the method comprising: generating a meshrepresentation of the physical entity, wherein the mesh representationcomprises a plurality of mesh elements; computing a linear system matrixA of coefficients by computing interactions between simple functionsdefined over sets of mesh elements; partitioning the mesh representationinto a plurality of partitions separated by partition boundaries,wherein each partition comprises at least one mesh element of theplurality of mesh elements, wherein at least one partition comprises atleast two mesh elements of the plurality of mesh elements; computing,using at least the plurality of partitions, a preconditioner for thelinear system matrix A, wherein the preconditioner is compatible withthe linear system of equations and provides at least basis functionsupport over at least two mesh elements, where coupling of thepreconditioner between partitions is only through basis functions at thepartition boundaries; using at least the linear system matrix A and thepreconditioner, determining an approximate numerical solution of thelinear system of equations, wherein the approximate numerical solutioncomprises the information indicative of at least one property of thephysical entity, wherein the at least one property comprises a fluidmechanical property; and outputting the approximate numerical solution.2. A method as in claim 1, where the preconditioner is itself a validsolution to a same set of physical equations that govern the full linearsystem of equations.
 3. A method as in claim 1, where computing apreconditioner operates to compute a preconditioning matrix K wherepartition boundaries are constrained to coincide with the edges of meshelements, and to compute mesh element interactions using reducedcoupling.
 4. A method as in claim 3, where mesh element interactionsbetween basis functions are computed only for half functions within thesame partition, where a half function denotes the function over any oneof multiple mesh elements for which it is defined, and where theinteractions of basis functions crossing a partition boundary arecomputed separately for each of the half functions such that nointeractions exist between basis function halves that are defined inseparate ones of the partitions, and those basis functions completelywithin a partition, referred to as interior elements, interact only withother interior elements and with boundary element halves within the samepartition.
 5. A method as in claim 4, further comprising sorting indicesof basis functions in the matrices A and K so that all internal elementsappear first, grouped according to their respective partitions, followedby all boundary elements, and where a resulting preconditioning matrix Kfor n partitions has the form: ${K = \begin{bmatrix}\lbrack {Ka}_{1} \rbrack & \; & \; & \; & \; \\\; & \lbrack {Ka}_{2} \rbrack & \; & \; & \; \\\; & \; & ⋰ & \; & \lbrack{Kb}\rbrack \\\; & \; & \; & \lbrack {Ka}_{n} \rbrack & \; \\\; & \lbrack{Kc}\rbrack & \; & \; & \lbrack{Kd}\rbrack\end{bmatrix}},$ where the sub matrix Ka is the block diagonal matrixcreated by the union of the matrices of internal element interactionsKa₁ through Ka_(n), Kd represents the interactions between the boundaryelements, and Kb and Kc are the interactions between the internal andboundary elements.
 6. A method as in claim 5, wherein determining anapproximate numerical solution further comprises iteratively solving asystem of equations Ax=f using the linear system matrix A, a vector f ofboundary conditions on each mesh element and the preconditioner matrix Kto provide an approximate solution x.
 7. A method as in claim 6, wherethe linear system matrix A is partitioned in the same manner as thepreconditioner using the same partitions, separate partitions, or acombination of the same and separate partitions.
 8. A method as in claim6, where iteratively solving comprises operating a conjugate gradientiterative solver.
 9. A method to determine information indicative of atleast one property of a physical entity by utilizing a linear system ofequations to represent the physical entity, the method comprising:generating a mesh representation of the physical entity, wherein themesh representation comprises a plurality of mesh elements; computing alinear system matrix A of coefficients by computing interactions betweensimple functions defined over sets of mesh elements; partitioning themesh representation into a plurality of partitions separated bypartition boundaries, wherein each partition comprises at least one meshelement of the plurality of mesh elements, wherein at least onepartition comprises at least two mesh elements of the plurality of meshelements; computing, using at least the plurality of partitions, apreconditioner for the linear system matrix A, wherein thepreconditioner is compatible with the linear system of equations andprovides at least basis function support over at least two meshelements, where coupling of the preconditioner between partitions isonly through basis functions at the partition boundaries; using at leastthe linear system matrix A and the preconditioner, determining anapproximate numerical solution of the linear system of equations,wherein the approximate numerical solution comprises the informationindicative of at least one property of the physical entity, wherein theat least one property comprises an acoustical property; and outputtingthe approximate numerical solution.
 10. A method as in claim 9, wherethe preconditioner is itself a valid solution to a same set of physicalequations that govern the full linear system of equations.
 11. A methodas in claim 9, where computing a preconditioner operates to compute apreconditioning matrix K where partition boundaries are constrained tocoincide with the edges of mesh elements, and to compute mesh elementinteractions using reduced coupling.
 12. A method as in claim 9, wheremesh element interactions between basis functions are computed only forhalf functions within the same partition, where a half function denotesthe function over any one of multiple mesh elements for which it isdefined, and where the interactions of basis functions crossing apartition boundary are computed separately for each of the halffunctions such that no interactions exist between basis function halvesthat are defined in separate ones of the partitions, and those basisfunctions completely within a partition, referred to as interiorelements, interact only with other interior elements and with boundaryelement halves within the same partition.
 13. A method as in claim 9,further comprising sorting indices of basis functions in the matrices Aand K so that all internal elements appear first, grouped according totheir respective partitions, followed by all boundary elements, andwhere a resulting preconditioning matrix K for n partitions has theform: ${K = \begin{bmatrix}\lbrack {Ka}_{1} \rbrack & \; & \; & \; & \; \\\; & \lbrack {Ka}_{2} \rbrack & \; & \; & \; \\\; & \; & ⋰ & \; & \lbrack{Kb}\rbrack \\\; & \; & \; & \lbrack {Ka}_{n} \rbrack & \; \\\; & \lbrack{Kc}\rbrack & \; & \; & \lbrack{Kd}\rbrack\end{bmatrix}},$ where the sub matrix Ka is the block diagonal matrixcreated by the union of the matrices of internal element interactionsKa₁ through Ka_(n), Kd represents the interactions between the boundaryelements, and Kb and Kc are the interactions between the internal andboundary elements.
 14. A method as in claim 13, wherein determining anapproximate numerical solution further comprises iteratively solving asystem of equations Ax=f using the linear system matrix A, a vector f ofboundary conditions on each mesh element and the preconditioner matrix Kto provide an approximate solution x.
 15. A method to determineinformation indicative of at least one property of a physical entity byutilizing a linear system of equations to represent the physical entity,the method comprising: generating a mesh representation of the physicalentity, wherein the mesh representation comprises a plurality of meshelements; computing a linear system matrix A of coefficients bycomputing interactions between simple functions defined over sets ofmesh elements; partitioning the mesh representation into a plurality ofpartitions separated by partition boundaries, wherein each partitioncomprises at least one mesh element of the plurality of mesh elements,wherein at least one partition comprises at least two mesh elements ofthe plurality of mesh elements; computing, using at least the pluralityof partitions, a preconditioner for the linear system matrix A, whereinthe preconditioner is compatible with the linear system of equations andprovides at least basis function support over at least two meshelements, where coupling of the preconditioner between partitions isonly through basis functions at the partition boundaries; using at leastthe linear system matrix A and the preconditioner, determining anapproximate numerical solution of the linear system of equations,wherein the approximate numerical solution comprises the informationindicative of at least one property of the physical entity, wherein theat least one property comprises a field scattering property of aradar-related component; and outputting the approximate numericalsolution.
 16. A method as in claim 15, where the preconditioner isitself a valid solution to a same set of physical equations that governthe full linear system of equations.
 17. A method as in claim 15, wherecomputing a preconditioner operates to compute a preconditioning matrixK where partition boundaries are constrained to coincide with the edgesof mesh elements, and to compute mesh element interactions using reducedcoupling.
 18. A method as in claim 17, where mesh element interactionsbetween basis functions are computed only for half functions within thesame partition, where a half function denotes the function over any oneof multiple mesh elements for which it is defined, and where theinteractions of basis functions crossing a partition boundary arecomputed separately for each of the half functions such that nointeractions exist between basis function halves that are defined inseparate ones of the partitions, and those basis functions completelywithin a partition, referred to as interior elements, interact only withother interior elements and with boundary element halves within the samepartition.
 19. A method as in claim 15, further comprising sortingindices of basis functions in the matrices A and K so that all internalelements appear first, grouped according to their respective partitions,followed by all boundary elements, and where a resulting preconditioningmatrix K for n partitions has the form: ${K = \begin{bmatrix}\lbrack {Ka}_{1} \rbrack & \; & \; & \; & \; \\\; & \lbrack {Ka}_{2} \rbrack & \; & \; & \; \\\; & \; & ⋰ & \; & \lbrack{Kb}\rbrack \\\; & \; & \; & \lbrack {Ka}_{n} \rbrack & \; \\\; & \lbrack{Kc}\rbrack & \; & \; & \lbrack{Kd}\rbrack\end{bmatrix}},$ where the sub matrix Ka is the block diagonal matrixcreated by the union of the matrices of internal element interactionsKa₁ through Ka_(n), Kd represents the interactions between the boundaryelements, and Kb and Kc are the interactions between the internal andboundary elements.
 20. A method to determine information indicative ofat least one property of a physical entity by utilizing a linear systemof equations to represent the physical entity, the method comprising:generating a mesh representation of the physical entity, wherein themesh representation comprises a plurality of mesh elements; computing alinear system matrix A of coefficients by computing interactions betweensimple functions defined over sets of mesh elements; partitioning themesh representation into a plurality of partitions separated bypartition boundaries, wherein each partition comprises at least one meshelement of the plurality of mesh elements, wherein at least onepartition comprises at least two mesh elements of the plurality of meshelements; computing, using at least the plurality of partitions, apreconditioner for the linear system matrix A, wherein thepreconditioner is compatible with the linear system of equations andprovides at least basis function support over at least two meshelements, where coupling of the preconditioner between partitions isonly through basis functions at the partition boundaries; using at leastthe linear system matrix A and the preconditioner, determining anapproximate numerical solution of the linear system of equations,wherein the approximate numerical solution comprises the informationindicative of at least one property of the physical entity, wherein theat least one property comprises an electromagnetic property of a printedcircuit board; and outputting the approximate numerical solution.
 21. Amethod as in claim 20, where the preconditioner is itself a validsolution to a same set of physical equations that govern the full linearsystem of equations.
 22. A method as in claim 20, where computing apreconditioner operates to compute a preconditioning matrix K wherepartition boundaries are constrained to coincide with the edges of meshelements, and to compute mesh element interactions using reducedcoupling.
 23. A method as in claim 22, where mesh element interactionsbetween basis functions are computed only for half functions within thesame partition, where a half function denotes the function over any oneof multiple mesh elements for which it is defined, and where theinteractions of basis functions crossing a partition boundary arecomputed separately for each of the half functions such that nointeractions exist between basis function halves that are defined inseparate ones of the partitions, and those basis functions completelywithin a partition, referred to as interior elements, interact only withother interior elements and with boundary element halves within the samepartition.
 24. A method as in claim 20, further comprising sortingindices of basis functions in the matrices A and K so that all internalelements appear first, grouped according to their respective partitions,followed by all boundary elements, and where a resulting preconditioningmatrix K for n partitions has the form: ${K = \begin{bmatrix}\lbrack {Ka}_{1} \rbrack & \; & \; & \; & \; \\\; & \lbrack {Ka}_{2} \rbrack & \; & \; & \; \\\; & \; & ⋰ & \; & \lbrack{Kb}\rbrack \\\; & \; & \; & \lbrack {Ka}_{n} \rbrack & \; \\\; & \lbrack{Kc}\rbrack & \; & \; & \lbrack{Kd}\rbrack\end{bmatrix}},$ where the sub matrix Ka is the block diagonal matrixcreated by the union of the matrices of internal element interactionsKa₁ through Ka_(n), Kd represents the interactions between the boundaryelements, and Kb and Kc are the interactions between the internal andboundary elements.